Question

Let (x1,x2,x3) and (y1,y2,y3) represent the coordinates with respect to the bases
B1 = {(1, 0, 0),(0, 1, 0),(0, 0, 1)}, B2 = {(1, 0, 0),(0, 1, 2),(0, 2, 1)}. If
Q(X) = 2x1^2+ 2x1x2 − 2x2x3 +x2^2+x3^2, find the representation of Q in terms of
(y1,y2,y3).

Answer 1

Answer on Question #62080 – Math – Linear Algebra

Question

Let $(x_1, x_2, x_3)$ and $(y_1, y_2, y_3)$ represent the coordinates with respect to the bases

$$B_1 = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}, \quad B_2 = \{(1, 0, 0), (0, 1, 2), (0, 2, 1)\}.$$

If

$$Q(X) = 2x_1^2 + 2x_1x_2 - 2x_2x_3 + x_2^2 + x_3^2,$$

find the representation of $Q$ in terms of $(y_1, y_2, y_3)$.

Solution

First of all, let us find the transition matrix $C = C_{B_1 \to B_2}$.

The following equalities are true:

$$(1, 0, 0) = 1 \cdot (1, 0, 0) + 0 \cdot (0, 1, 0) + 0 \cdot (0, 0, 1);$$ $$(0, 1, 2) = 0 \cdot (1, 0, 0) + 1 \cdot (0, 1, 0) + 2 \cdot (0, 0, 1);$$ $$(0, 2, 1) = 0 \cdot (1, 0, 0) + 2 \cdot (0, 1, 0) + 1 \cdot (0, 0, 1).$$

So

$$C = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 1 \end{pmatrix}.$$

Given $Q(X) = 2x_1^2 + 2x_1x_2 - 2x_2x_3 + x_2^2 + x_3^2$, the matrix of $Q(X)$ in the base $B_1$:

$$A = \begin{pmatrix} 2 & 1 & 0 \\ 1 & 1 & -1 \\ 0 & -1 & 1 \end{pmatrix}.$$

Then the matrix of the form $Q$ in the base $B_2$ is equal to

$$B = C^T A C = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 1 \end{pmatrix} \begin{pmatrix} 2 & 1 & 0 \\ 1 & 1 & -1 \\ 0 & -1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 1 & 2 \\ 1 & 1 & -1 \\ 2 & -1 & 1 \end{pmatrix}.$$

So the form $Q$ in the base $B_2$ will be

$$Q(Y) = 2y_1^2 + y_2^2 + y_3^2 + 2y_1y_2 + 4y_1y_3 - 2y_2y_3.$$

Answer: $Q(Y) = 2y_1^2 + y_2^2 + y_3^2 + 2y_1y_2 + 4y_1y_3 - 2y_2y_3$.

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